The first column \(\mathbf {a}_1\)  of \(\mathbf {\tilde{A}}\)  may be separated as \(\mathbf {\tilde{A}}=[\mathbf {a}_1 \hspace{3.0pt} \mathbf {A}]\) . This yields the reduced branch-bus incidence matrix \(\mathbf {A}\)  of \(\mathcal {G}\) . An efficient model for imposing graph connectivity put forth in [1]} posits that a graph with vertex set \(\mathcal {V}\)  and reduced branch-bus incidence matrix \(\mathbf {A}\) , is connected if and only if there exists a vector \(\mathbf {f}\in {R}^{|\mathcal {E}|}\) , such that \(\mathbf {A}^T\mathbf {f}=\mathbf {1}\) . For proof, see [1]}. For a physical interpretation, consider every vertex in \(\mathcal {V}\setminus \lbrace 1\rbrace \)  injects unit virtual commodity into the network represented by the graph. Then, \(\mathbf {f}\)  denotes the flow of commodities on the edges. If this flow setup is feasible, then there must be a withdrawal of \(|\mathcal {V}|-1\)  units at vertex 1, and every vertex in \(\mathcal {V}\setminus \lbrace 1\rbrace \)  must have a path to reach vertex 1. Vertex 1 may be arbitrarily chosen.
Stating a well-known result from graph theory, a tree with \(n\)  vertices has exactly \(n-1\)  edges. Hence, the radiality constraints become:
\(\mathbf {A}^T\mathbf {f}&=\mathbf {1} \\-(|\mathcal {V}|-1)\theta \le & \text{ }\mathbf {f}\le (|\mathcal {V}|-1)\theta ,\\\theta &\in \lbrace 0,1\rbrace ^{|\mathcal {E}|} \\\mathbf {1}^T\theta &=|\mathcal {V}|-1\\\mathbf {b^e}&\le \theta \) 
